The general solution is of the form: [itex]y(x)=\sum_i^m c_ix^{(m_i)}[/itex]
where [itex] m_i[/itex] are the roots of the polynomial when [itex]x^m[/itex] is substituted into the ODE.

The polynomial equation for the ODE of the original posts is:

[tex]4m^2+4m-3=0[/tex]

Thus, the solution is seen to be:

[tex] y(x)=c_1x^{\frac{1}{2}}+c_2x^{\frac{-3}{2}}[/tex]

Can you attempt to solve the following:

[tex] 4x^2y''+8xy'+3y=0[/tex]

This produces complex roots which necessarily involve terms of the form:
[itex]\cos(\ln[u(x)])[/itex] and [itex]\sin(\ln[v(x)])[/tex] in which [itex]u(x)[/itex] and [itex]v(x)[/itex] are powers of x. Just absorb the imaginary i into the arbritrary constants to get a real-valued function. Although I believe the function is valid only for x>0 because of the logarithm involved. This is what I got:
[tex]
y[x]={c_1}\multsp \frac{\cos \big[\ln \big[{x^{\frac{1}{{\sqrt{2}}}}}\big]\big]}{{\sqrt{x}}}+\\
\noalign{\vspace{2.125ex}}
\hspace{2.em} {c_2}\multsp \frac{\sin \big[\ln \big[{x^{\frac{1}{{\sqrt{2}}}}}\big]\big]}{{\sqrt{x}}}
[/tex]